Reference answers to semi-final questions (group 1)

Fill in the blanks (each question 10, ***80)

Title 1 2 3 4 5 6 7 8

Answer 97 48 57 85

Second, briefly answer the following questions (each question 10, * * * 40, requires a brief process)

9. solution: let x, y, z and t be integers, and suppose that

( 1)

Comparing the coefficients of formulas A, B and C above, there should be

(2)

Here, you can have it.,,, You have it

(3)

Because sum can be divisible by 13, it can be divisible by 13.

Explain that the table is the algebraic sum of two algebraic expressions divisible by 13, and the expressions are not unique. For example, you take it, you have it, you have it.

In fact, (2) is a set of binary indefinite equations with integral coefficients. Let's solve the first problem first

Where k is an arbitrary integer,

Will be substituted into the rest of the equation, then the solution is

Where k is an arbitrary integer,

Then there can be

Scoring reference: There is an algebraic expression similar to (3), giving a score of 10.

10, solution: As shown in Figure 3a, AE, EN and NC are connected together, which is easy to know.

By,

The above two formulas are added together.

( 1)

Area of quadrilateral AECN =

Connect AC, as shown in Figure 3b, which is easy to know from the triangle area formula.

,

The above two formulas are added together.

Quadrilateral area AECN = (2)

Add (1) and (2) to obtain.

,

because

,

So figure 3b

, 。

A:

Scoring reference: ① The result (1) can be deduced by the triangle area formula, giving 4 points; ② The result (2) can be derived by using the triangle area formula, giving 4 points; ③ Give the correct answer and give 2 points.

1 1. Solution: The method of filling numbers is the exclusion method, and (m, n) is used to represent the square located in the m-th row and the n-th column.

There are 9' s in the seventh, eighth and third rows, so the 9' s in the "Little Nine Palace" in the lower left corner of the original figure 4 should be filled in the (9,2) grid figure 4a.

Zizhong; 1 column has the number 5 in the seventh row of the second column, so the number 5 in the "Little Nine Palaces" box in the lower right corner of Figure 4 can only be filled with (9,3); The seventh and eighth lines have the number 6, and the number 6 in the "Little Nine Palace" at the bottom of Figure 4 should be filled in (9, 6); At this point, the numbers 7 and 3 are still missing in the ninth row. Because the number 7 is in the ninth column, you should fill in 7(9, 8); 3 Fill in (9, 9) naturally, and the filling method is shown in Figure 4a.

These nine numbers are 495 186273.

Scoring reference: ① Give the correct answer and give 5 points; (2) For the filling method of the five spaces of "Xiao Jiu Gong" in the left of Figure 4, if the reasons can be explained, 5 points will be given, and each space will be given 1 minute; (3) Even if the final answer is incorrect, the correct fill-in-the-blank method should be given an appropriate score;

12, answer:

(1) First, choose 1 from six points to be a vertex of the triangle, and there are six ways to take it; Then choose 1 as the second vertex of the triangle from the remaining five points, and there are five ways to take it; Then choose 1 from the remaining four points as the third vertex of the triangle. There are four ways to take it. Because any three points are not on a straight line, the three points selected in this way can be made into 1 triangle. However, if the selected three points are the same, the triangle is the same, and there are 3×2× 1=6 ways to take the same three points. Therefore, these six points can be used as vertices to construct a different triangle.

(2) Each triangle has three vertices, so six points can only construct two triangles with no common vertices at most.

(3) Write down these six points with English capital letters A, B, C, D, E and F. Suppose you can choose five triangles with no common edges. They have 15 vertices and need 15 English capital letters. There are only six different English capital letters here. Therefore, at least three of these five triangles have the same vertex, so it is just as well to set it as A. According to the assumption, these three triangles have no common edge, that is, except the common vertex A, the other six vertices are different from each other, that is, the letters representing these six vertices are different. However, apart from A, we only have five different letters. Therefore, it is impossible to have five triangles, and every two have no common side.

Obviously, these four triangles have no common edge. So you can choose up to four triangles, and no two of them have the same side.

Scoring reference: ① Give 3 points for answering the first question correctly; 2 Answer the second question correctly and give 2 points; (3) In the third question, 2 points will be given if the answer is correct, and 2 points will be given if the reason can be explained.

Three. Answer the following questions (15 for each question, * * * 30 for each question, which requires detailed procedures).

13, solution: let Liu Fang be x years old.

(1) The sum of the ages of Liu Fang and Lulu is 36 years old, which is even, and their age difference is even, while the age difference between Lulu and her mother is odd, so Lulu's mother is not Liu Fang. I noticed that Feifei is 29 years younger than Liu Fang. Feifei's mother is not Liu Fang, so her strong mother is Liu Fang.

② The sum of the ages of Zhuang Zhuang and his mother Liu Fang is (

Lulu is old, so should his mother, right?

Feifei is years old, her mother should be years old, and for

Because all six people are *** 105 years old, so,

.

③ x=32, Feifei is 29 years younger than Liu Fang, so Feifei is 3 years old; The sum of the ages of Lulu and Liu Fang is 36, and Lulu is 4 years old. The sum of the ages of Lulu and Wang Xue is 35 years old, so Wang Xue is 3 1 year old.

Wang Xue is Lulu's mother; Zhuang Zhuang is 5 years old, Feifei is 3 years old and Lulu is 4 years old.

Scoring reference: ① In the first step, it can be judged that the strong mother is Liu Fang, giving 5 points; (2) Who is Lulu's mother who can answer correctly, give 5 points; (3) Can correctly answer the questions of children under three years old, give 5 points.

14, solution:

(1) because, yes. So it can be expressed as the sum of the reciprocal of three mutually different positive integers (the expression is not unique).

(2) We assume that the problem now is to find the whole A, B and C and satisfy them.

By, yes.

, therefore,

So ... again, so, so it's still 16.

If so, yes, because, besides,

therefore ...

Therefore, 100 or 12 1. Substitution is 100 and 12 1 respectively, and neither of them is a complete square number, indicating that there is no solution.

If, then. Similarly, the following can be obtained:

, that is,

At this point, it is not an integer.

To sum up, it cannot be expressed as the sum of the reciprocal of three different complete squares.

Scoring reference: ① Give 5 points for answering the first question correctly (the answer is not unique); (2) If you can get or 16, give 6 points; (3) Can we discuss whether 16 and 16 can be expressed as the sum of the reciprocal of three different complete squares, and give 2 points ***4 points each; (4) Give appropriate scores to the reasonable and correct derivation of algebraic expressions.

Special note: As the answer to each question may not be unique, the above answers and scores are for reference only.